x (x)^ 3 2 (x)^ 3 2 d x - Vidyadip

Question

$\int \sqrt{x} \sec (x)^{\frac{3}{2}} \tan (x)^{\frac{3}{2}} d x$

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Answer

Let $I =\int \sqrt{x} \cdot \sec \left(x^{\frac{3}{2}}\right) \cdot \tan \left(x^{\frac{3}{2}}\right) d x$
Put $x^{\frac{3}{2}}= t$
$ \therefore \frac{3}{2} x^{\frac{1}{2}} d x= dt$
$\therefore \sqrt{x} d x=\frac{2}{3} dt$
$\therefore I =\frac{2}{3} \int sec t \cdot tan t \cdot dt$
$=\frac{2}{3} \sec t + c$
$\therefore I =\frac{2}{3} \sec \left(x^{\frac{3}{2}}\right)+ c $

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